Complex geodesics, their boundary regularity, and a Hardy-Littlewood-type lemma

Abstract

Abstract. We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to ∂D. This example suggests that continuity at the boundary of the complex geodesics of a convex domain Ω ⋐ C n , n ≥ 2, is affected by the extent to which ∂Ω curves or bends at each boundary point. We provide a sufficient condition to this effect (on C 1 -smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy-Littlewood-type lemma that might be of independent interest.

Goldilocks domains, a weak notion of visibility, and applications

“…One needs a device to quantify how flat ∂Ω can get at smooth points. This is accomplished by the notion of the support of Ω from the outside, which was introduced in [Bha16]. The following definition has been adapted from [Bha16] -which focuses on domains with C 1 -smooth boundaryto admit convex domains with non-smooth boundaries as well.…”

Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

“…Following the ideas presented in [20] and [21] it is probable that a good tool for providing some sufficient conditions for the Gromov hyperbolicity would be the analysis of properties of real geodesics with respect to the Kobayashi distance. The problem of regularity of complex geodesics is also a very important one and may be used in many problems of complex analysis -one may look for instance at papers [20], [21], [3], [2] to name just a few that have appeared recently. The very recent papers of Zajac allow us to provide many strong regularity properties of complex geodesics in convex tube domains which also show that real geodesics are in fact much more regular (for tube domains with bounded smooth bases) than in the situation studied by Zimmer.…”

On the continuous extension of Kobayashi isometries

“…We now state the result from [2] alluded to above. To state it, we need, given a bounded convex domain Ω ⊂ C n with C 1 -smooth boundary, the notion of a function that supports Ω from the outside.…”

“…The earliest result in this direction was given by Lempert [8], which states that if Ω ⊂ C n is strongly convex with C k -smooth boundary, k 2, then every complex geodesic F : D → Ω extends to a C k−2 -smooth mapping on D (by a C 0 -smooth mapping we mean a continuous one). Since then, there has been a number of works dealing with the continuous (or smooth) extension of complex geodesics; see [1,10,2,13].…”